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In this paper, we introduce a new technique for measuring the hypothetical non-monotonicity of the argument of the vector tracing the omitted arc of a support point of the class
. It has been previously shown that the number of sign changes of (arg w(t))′ on the omitted arc is finite....
We study the zero sequences of the non-trivial solutions of
$$f'' + A(z)f = 0,$$
, where A(z) is analytic in the unit disc. Namely, we consider the following two problems:
find a growth condition on A(z) such that the zero sequence of any non-trivial solution of (*) is a Blaschke...
0, …, a
k−1 be analytic functions on a domain Ω. Let F be a family of meromorphic functions f defined on Ω such that f ≠ 0 and f
(k) + a
(k−1) + … + a
f ≠ 0 on Ω, for all f ∊ F. Then f′ / f: f ∊ F is a normal family. Furthermore, let a
k−1 be meromorphic functions on a...
We consider finite difference equations of Vekua type. Main goal of the paper is to prove a representation formula for the solution of homogeneous equations in the form of a product with one factor being a discrete holomorphic function.
Let Ω be a bounded open and connected subset of ℝm which has a C
∞-boundary Σ and let F
k ∊ C
∞(Σ) be a k-multi-vector valued function on Σ. Under which conditions can F
k be decomposed as F
k = F
+ + F
k− where F
+- are extendable to harmonic k-multi-vector fields in Ω± with Ω+ = Ω and...
We consider complex polynomials of degree n that are bounded by one in the unit disc and give estimates on the size of the radius R
n of the disc where the sum of the moduli of the individual terms of the polynomial is less than one. We find that there are positive constants C
2 such that...
We improve a result of Anderson, Baker and Clunie on exceptional sets for the differential polynomial f′ f, when f is a transcendental entire function.
Let G be a n-connected domain in the complex plane and let z
0 ∈ G be fixed. We consider the class H
0) of all holomorphic functions f of G into the unit disk E with f(z
0) = 0. For z ∈ G
0 the Carathéodory distance c(z, z
0) is defined as the maximum of ¦f(z)¦, f ∈ H
Let G be a bounded Jordan domain in ℂ and let w n = 0 be an analytic function on G such that tS
dm < ∞, where dm is the area measure. We investigate the zero distribution of the sequence of polynomials that are orthogonal on G with respect to ¦ω¦2
dm. We find that such a distribution...
On a Stein manifold, we obtain generic versions of two covering theorems of Valiron and Birkhoff respectively. Namely, we show that most equidimensional holomorphic mappings into complex Euclidean space contain arbitrarily large ‘schlicht’ balls in their images. Moreover, most global holomorphic...
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